// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2010 Vincent Lejeune
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_QR_H
#define EIGEN_QR_H

namespace Eigen {

namespace internal {
    template <typename _MatrixType> struct traits<HouseholderQR<_MatrixType>> : traits<_MatrixType>
    {
        typedef MatrixXpr XprKind;
        typedef SolverStorage StorageKind;
        typedef int StorageIndex;
        enum
        {
            Flags = 0
        };
    };

}  // end namespace internal

/** \ingroup QR_Module
  *
  *
  * \class HouseholderQR
  *
  * \brief Householder QR decomposition of a matrix
  *
  * \tparam _MatrixType the type of the matrix of which we are computing the QR decomposition
  *
  * This class performs a QR decomposition of a matrix \b A into matrices \b Q and \b R
  * such that 
  * \f[
  *  \mathbf{A} = \mathbf{Q} \, \mathbf{R}
  * \f]
  * by using Householder transformations. Here, \b Q a unitary matrix and \b R an upper triangular matrix.
  * The result is stored in a compact way compatible with LAPACK.
  *
  * Note that no pivoting is performed. This is \b not a rank-revealing decomposition.
  * If you want that feature, use FullPivHouseholderQR or ColPivHouseholderQR instead.
  *
  * This Householder QR decomposition is faster, but less numerically stable and less feature-full than
  * FullPivHouseholderQR or ColPivHouseholderQR.
  *
  * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
  *
  * \sa MatrixBase::householderQr()
  */
template <typename _MatrixType> class HouseholderQR : public SolverBase<HouseholderQR<_MatrixType>>
{
public:
    typedef _MatrixType MatrixType;
    typedef SolverBase<HouseholderQR> Base;
    friend class SolverBase<HouseholderQR>;

    EIGEN_GENERIC_PUBLIC_INTERFACE(HouseholderQR)
    enum
    {
        MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
        MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };
    typedef Matrix<Scalar,
                   RowsAtCompileTime,
                   RowsAtCompileTime,
                   (MatrixType::Flags & RowMajorBit) ? RowMajor : ColMajor,
                   MaxRowsAtCompileTime,
                   MaxRowsAtCompileTime>
        MatrixQType;
    typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
    typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
    typedef HouseholderSequence<MatrixType, typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type> HouseholderSequenceType;

    /**
      * \brief Default Constructor.
      *
      * The default constructor is useful in cases in which the user intends to
      * perform decompositions via HouseholderQR::compute(const MatrixType&).
      */
    HouseholderQR() : m_qr(), m_hCoeffs(), m_temp(), m_isInitialized(false) {}

    /** \brief Default Constructor with memory preallocation
      *
      * Like the default constructor but with preallocation of the internal data
      * according to the specified problem \a size.
      * \sa HouseholderQR()
      */
    HouseholderQR(Index rows, Index cols) : m_qr(rows, cols), m_hCoeffs((std::min)(rows, cols)), m_temp(cols), m_isInitialized(false) {}

    /** \brief Constructs a QR factorization from a given matrix
      *
      * This constructor computes the QR factorization of the matrix \a matrix by calling
      * the method compute(). It is a short cut for:
      * 
      * \code
      * HouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
      * qr.compute(matrix);
      * \endcode
      * 
      * \sa compute()
      */
    template <typename InputType>
    explicit HouseholderQR(const EigenBase<InputType>& matrix)
        : m_qr(matrix.rows(), matrix.cols()), m_hCoeffs((std::min)(matrix.rows(), matrix.cols())), m_temp(matrix.cols()), m_isInitialized(false)
    {
        compute(matrix.derived());
    }

    /** \brief Constructs a QR factorization from a given matrix
      *
      * This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when
      * \c MatrixType is a Eigen::Ref.
      *
      * \sa HouseholderQR(const EigenBase&)
      */
    template <typename InputType>
    explicit HouseholderQR(EigenBase<InputType>& matrix)
        : m_qr(matrix.derived()), m_hCoeffs((std::min)(matrix.rows(), matrix.cols())), m_temp(matrix.cols()), m_isInitialized(false)
    {
        computeInPlace();
    }

#ifdef EIGEN_PARSED_BY_DOXYGEN
    /** This method finds a solution x to the equation Ax=b, where A is the matrix of which
      * *this is the QR decomposition, if any exists.
      *
      * \param b the right-hand-side of the equation to solve.
      *
      * \returns a solution.
      *
      * \note_about_checking_solutions
      *
      * \note_about_arbitrary_choice_of_solution
      *
      * Example: \include HouseholderQR_solve.cpp
      * Output: \verbinclude HouseholderQR_solve.out
      */
    template <typename Rhs> inline const Solve<HouseholderQR, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif

    /** This method returns an expression of the unitary matrix Q as a sequence of Householder transformations.
      *
      * The returned expression can directly be used to perform matrix products. It can also be assigned to a dense Matrix object.
      * Here is an example showing how to recover the full or thin matrix Q, as well as how to perform matrix products using operator*:
      *
      * Example: \include HouseholderQR_householderQ.cpp
      * Output: \verbinclude HouseholderQR_householderQ.out
      */
    HouseholderSequenceType householderQ() const
    {
        eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
        return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
    }

    /** \returns a reference to the matrix where the Householder QR decomposition is stored
      * in a LAPACK-compatible way.
      */
    const MatrixType& matrixQR() const
    {
        eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
        return m_qr;
    }

    template <typename InputType> HouseholderQR& compute(const EigenBase<InputType>& matrix)
    {
        m_qr = matrix.derived();
        computeInPlace();
        return *this;
    }

    /** \returns the absolute value of the determinant of the matrix of which
      * *this is the QR decomposition. It has only linear complexity
      * (that is, O(n) where n is the dimension of the square matrix)
      * as the QR decomposition has already been computed.
      *
      * \note This is only for square matrices.
      *
      * \warning a determinant can be very big or small, so for matrices
      * of large enough dimension, there is a risk of overflow/underflow.
      * One way to work around that is to use logAbsDeterminant() instead.
      *
      * \sa logAbsDeterminant(), MatrixBase::determinant()
      */
    typename MatrixType::RealScalar absDeterminant() const;

    /** \returns the natural log of the absolute value of the determinant of the matrix of which
      * *this is the QR decomposition. It has only linear complexity
      * (that is, O(n) where n is the dimension of the square matrix)
      * as the QR decomposition has already been computed.
      *
      * \note This is only for square matrices.
      *
      * \note This method is useful to work around the risk of overflow/underflow that's inherent
      * to determinant computation.
      *
      * \sa absDeterminant(), MatrixBase::determinant()
      */
    typename MatrixType::RealScalar logAbsDeterminant() const;

    inline Index rows() const { return m_qr.rows(); }
    inline Index cols() const { return m_qr.cols(); }

    /** \returns a const reference to the vector of Householder coefficients used to represent the factor \c Q.
      * 
      * For advanced uses only.
      */
    const HCoeffsType& hCoeffs() const { return m_hCoeffs; }

#ifndef EIGEN_PARSED_BY_DOXYGEN
    template <typename RhsType, typename DstType> void _solve_impl(const RhsType& rhs, DstType& dst) const;

    template <bool Conjugate, typename RhsType, typename DstType> void _solve_impl_transposed(const RhsType& rhs, DstType& dst) const;
#endif

protected:
    static void check_template_parameters() { EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); }

    void computeInPlace();

    MatrixType m_qr;
    HCoeffsType m_hCoeffs;
    RowVectorType m_temp;
    bool m_isInitialized;
};

template <typename MatrixType> typename MatrixType::RealScalar HouseholderQR<MatrixType>::absDeterminant() const
{
    using std::abs;
    eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
    eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
    return abs(m_qr.diagonal().prod());
}

template <typename MatrixType> typename MatrixType::RealScalar HouseholderQR<MatrixType>::logAbsDeterminant() const
{
    eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
    eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
    return m_qr.diagonal().cwiseAbs().array().log().sum();
}

namespace internal {

    /** \internal */
    template <typename MatrixQR, typename HCoeffs>
    void householder_qr_inplace_unblocked(MatrixQR& mat, HCoeffs& hCoeffs, typename MatrixQR::Scalar* tempData = 0)
    {
        typedef typename MatrixQR::Scalar Scalar;
        typedef typename MatrixQR::RealScalar RealScalar;
        Index rows = mat.rows();
        Index cols = mat.cols();
        Index size = (std::min)(rows, cols);

        eigen_assert(hCoeffs.size() == size);

        typedef Matrix<Scalar, MatrixQR::ColsAtCompileTime, 1> TempType;
        TempType tempVector;
        if (tempData == 0)
        {
            tempVector.resize(cols);
            tempData = tempVector.data();
        }

        for (Index k = 0; k < size; ++k)
        {
            Index remainingRows = rows - k;
            Index remainingCols = cols - k - 1;

            RealScalar beta;
            mat.col(k).tail(remainingRows).makeHouseholderInPlace(hCoeffs.coeffRef(k), beta);
            mat.coeffRef(k, k) = beta;

            // apply H to remaining part of m_qr from the left
            mat.bottomRightCorner(remainingRows, remainingCols)
                .applyHouseholderOnTheLeft(mat.col(k).tail(remainingRows - 1), hCoeffs.coeffRef(k), tempData + k + 1);
        }
    }

    /** \internal */
    template <typename MatrixQR,
              typename HCoeffs,
              typename MatrixQRScalar = typename MatrixQR::Scalar,
              bool InnerStrideIsOne = (MatrixQR::InnerStrideAtCompileTime == 1 && HCoeffs::InnerStrideAtCompileTime == 1)>
    struct householder_qr_inplace_blocked
    {
        // This is specialized for LAPACK-supported Scalar types in HouseholderQR_LAPACKE.h
        static void run(MatrixQR& mat, HCoeffs& hCoeffs, Index maxBlockSize = 32, typename MatrixQR::Scalar* tempData = 0)
        {
            typedef typename MatrixQR::Scalar Scalar;
            typedef Block<MatrixQR, Dynamic, Dynamic> BlockType;

            Index rows = mat.rows();
            Index cols = mat.cols();
            Index size = (std::min)(rows, cols);

            typedef Matrix<Scalar, Dynamic, 1, ColMajor, MatrixQR::MaxColsAtCompileTime, 1> TempType;
            TempType tempVector;
            if (tempData == 0)
            {
                tempVector.resize(cols);
                tempData = tempVector.data();
            }

            Index blockSize = (std::min)(maxBlockSize, size);

            Index k = 0;
            for (k = 0; k < size; k += blockSize)
            {
                Index bs = (std::min)(size - k, blockSize);  // actual size of the block
                Index tcols = cols - k - bs;                 // trailing columns
                Index brows = rows - k;                      // rows of the block

                // partition the matrix:
                //        A00 | A01 | A02
                // mat  = A10 | A11 | A12
                //        A20 | A21 | A22
                // and performs the qr dec of [A11^T A12^T]^T
                // and update [A21^T A22^T]^T using level 3 operations.
                // Finally, the algorithm continue on A22

                BlockType A11_21 = mat.block(k, k, brows, bs);
                Block<HCoeffs, Dynamic, 1> hCoeffsSegment = hCoeffs.segment(k, bs);

                householder_qr_inplace_unblocked(A11_21, hCoeffsSegment, tempData);

                if (tcols)
                {
                    BlockType A21_22 = mat.block(k, k + bs, brows, tcols);
                    apply_block_householder_on_the_left(A21_22, A11_21, hCoeffsSegment, false);  // false == backward
                }
            }
        }
    };

}  // end namespace internal

#ifndef EIGEN_PARSED_BY_DOXYGEN
template <typename _MatrixType>
template <typename RhsType, typename DstType>
void HouseholderQR<_MatrixType>::_solve_impl(const RhsType& rhs, DstType& dst) const
{
    const Index rank = (std::min)(rows(), cols());

    typename RhsType::PlainObject c(rhs);

    c.applyOnTheLeft(householderQ().setLength(rank).adjoint());

    m_qr.topLeftCorner(rank, rank).template triangularView<Upper>().solveInPlace(c.topRows(rank));

    dst.topRows(rank) = c.topRows(rank);
    dst.bottomRows(cols() - rank).setZero();
}

template <typename _MatrixType>
template <bool Conjugate, typename RhsType, typename DstType>
void HouseholderQR<_MatrixType>::_solve_impl_transposed(const RhsType& rhs, DstType& dst) const
{
    const Index rank = (std::min)(rows(), cols());

    typename RhsType::PlainObject c(rhs);

    m_qr.topLeftCorner(rank, rank).template triangularView<Upper>().transpose().template conjugateIf<Conjugate>().solveInPlace(c.topRows(rank));

    dst.topRows(rank) = c.topRows(rank);
    dst.bottomRows(rows() - rank).setZero();

    dst.applyOnTheLeft(householderQ().setLength(rank).template conjugateIf<!Conjugate>());
}
#endif

/** Performs the QR factorization of the given matrix \a matrix. The result of
  * the factorization is stored into \c *this, and a reference to \c *this
  * is returned.
  *
  * \sa class HouseholderQR, HouseholderQR(const MatrixType&)
  */
template <typename MatrixType> void HouseholderQR<MatrixType>::computeInPlace()
{
    check_template_parameters();

    Index rows = m_qr.rows();
    Index cols = m_qr.cols();
    Index size = (std::min)(rows, cols);

    m_hCoeffs.resize(size);

    m_temp.resize(cols);

    internal::householder_qr_inplace_blocked<MatrixType, HCoeffsType>::run(m_qr, m_hCoeffs, 48, m_temp.data());

    m_isInitialized = true;
}

/** \return the Householder QR decomposition of \c *this.
  *
  * \sa class HouseholderQR
  */
template <typename Derived> const HouseholderQR<typename MatrixBase<Derived>::PlainObject> MatrixBase<Derived>::householderQr() const
{
    return HouseholderQR<PlainObject>(eval());
}

}  // end namespace Eigen

#endif  // EIGEN_QR_H
